Introduction

In the cold region of northern China, as an important agriculture facility, the greenhouses have played an irreplaceable role in planting crops, vegetables, flowers and raising livestock and poultry1. Greenhouse in China had experienced three stages. The first stage was from 1960 s to 1970 s, at that time facility agriculture gradually appeared in China. However, these plastic greenhouses are simple in structure and single in function. The second stage was from 1970 s to 1990 s, greenhouses have attracted much attention in China. The third stage was from 1990 s to now, greenhouses in China had become more and more modernized through the continuous efforts of scientific researchers and adopting foreign advanced technique in greenhouse control. Greenhouses in China can be grouped into glass greenhouses, solar greenhouses and plastic greenhouses according to materials and construction forms. For glass greenhouses, the covering material is glass, and this greenhouse has the advantage of high construction accuracy and long service life, but the cost is high. Solar greenhouses are composed of side enclosure wall, front roof and back roof. As the lighting surface of the greenhouse, the front roof is often made up of trusses or single steel pipes as the structure. The back roof was usually made of masonry structure or rigid skeleton insulation materials for heat preservation and storage. Plastic greenhouses were enclosed with plastic films to form a four-sided lighting structure, which is easy to construct with low cost and convenient to use, so they have been widely used in China. According to the analysis and statistics of the development of greenhouses in China in 2022, plastic greenhouses accounted for 61.52% of the total market share. Plastic greenhouses skeletons were originally made of bamboo, concrete and steel, and steel-skeleton was widely used currently. Steel skeleton plastic greenhouses can be grouped into truss type and single-pipe type as shown in Fig. 1.

Fig. 1
figure 1

Two types of steel skeleton plastic greenhouses.

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Although the plastic greenhouses were designed, constructed and built in the same way as civil engineering, its design and construction were not mandatory in comparison with building construction. Some simple plastic greenhouses were built by farmers themselves with low bearing capacity and poor durability. With the development of agricultural modernization and the increase of farmers’ income, more and more farmers tend to adopt plastic greenhouses with high bearing capacity, reliable safety, long service life, beautiful appearance, excellent performance cost ratio and uniform specifications. The FEPG satisfies all these requirements. The cross section of 30 mm× 60 mm× 2 mm greenhouse skeleton currently used in the market is shown in Fig. 2.

Fig. 2
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Flat-elliptical pipe and the cross section.

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Photograph of the FEPG is shown in Fig. 3.

Fig. 3
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Photograph of the FEPG.

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The flat-elliptical pipe is made of the closed rectangular steel pipe with the right angle corner made into circle. Compared with the rectangular steel pipe in the same size, the film of this greenhouse is not easy to be scratched because of the circle shape. Moreover, the outline area and the inertia moment along the force direction become bigger so the stiffness and the stability of the greenhouse increase. The surface of the flat-elliptical pipe is galvanized to enhance the corrosion resistance. Compared with the truss type greenhouse in the same mass, the stiffness and load-bearing capacity of the flat-elliptical pipe greenhouse have been significantly improved2.

Plastic greenhouses not only provide space for plant growing and animal breeding, but also withstand various loads imposed on it and resist extreme natural disasters. With the characteristics of lightweight, small stiffness and high flexibility, greenhouses are quite sensitive to wind loads. So, the wind loads, as one of the main control loads in the design of greenhouses, play a crucial role in ensuring the reliability of the structure. The plastic greenhouses are destroyed by strong winds, and these accidents occur each year and bring lots of financial loss even threaten people’s lives3. Therefore, wind and snow loads are the major factors in design of greenhouse skeleton structures, which takes a crucial effect for structural reliability.

At present, the structural design of greenhouses in China follows the Chinese standard GB/T51183-2016 and the Chinese standard GB50009-20124,5.

In the aspect of wind resistance design for greenhouse, scholars at home and abroad have done a lot of research. Ding, Li and Shi established the finite element model of truss plastic greenhouse in consideration of the influence of skin effects in the static analysis of the structure6,7. Kim, Lee, Yeo and Lee evaluated the structure’s safety of plastic greenhouse subjected to wind load based on five different greenhouse design standards8. Ren analyzed and evaluated the strength, stiffness and stability characteristics of the greenhouse by using the finite element method subjected to different combinations of static wind load and snow load9. Uematsu and Takahashi simulated the collapse process of a truss plastic greenhouse under static wind load considering the fluid-structure interaction effects10. After analyzing the wind-induced response of plastic greenhouse in time domain, Jiang, and Bai drew the conclusion: the dynamic responses of greenhouse structure are underestimated if the amplification effects of fluctuating wind loads are not considered11. Wang and Li used ANSYS finite element software to conduct a comparative analysis of static and dynamic time history on a solar greenhouse12,13,14. The variations of axial force and displacement of structure were obtained. The conclusion is that it is necessary to consider the dynamic characteristics of wind load in the wind resistant design for greenhouses15.

As it can be seen from the above research results, scholars focused on exploring the bearing capacity of the greenhouse structure based on the static analysis, while there were relatively few studies on the effects of fluctuating wind load on the greenhouse. While finite element analysis and the Linear Filtering Method have been established in the field, this study presents a novel application to the flat-elliptical pipe greenhouse (FEPG), a structure that has not been previously studied under fluctuating wind loads. The greenhouse is a typical wind-induced sensitive structure. According to current Chinese standards, the influence of fluctuating wind on the structure was not considered in the calculation formulas due to the small height of the greenhouse. This was the main reason why the greenhouse is still destroyed subjected to the local strong wind conforming to design code.

In this study, a finite element model of the FEPG was established by using ABAQUS software. The variation of stress and displacement of the FEPG under fluctuating wind was analyzed. The results of this study provide a scientific basis for the enhanced design and optimization of flat-elliptical pipe greenhouse (FEPG), promoting their use in agricultural applications.

Materials and methods

Finite element model

The plastic greenhouses in practical engineering are 3-D structures, but in the current Chinese code, the greenhouse is simplified as single skeleton in 2-D subjected to wind load. Although many scholars believe that 2-D structure is sufficient to substitute a spatial structure, 2-D structures cannot demonstrate the out-of-plane failure of greenhouse skeletons, which is the most common type of failure in practical greenhouse skeletons16,17. And 2-D structure cannot express the distribution of wind load along the longitudinal direction of the greenhouses. Therefore, a spatial model of a flat-elliptical pipe plastic greenhouse was established by using ABAQUS, in which the plastic film was added as a model unit. To obtain the structure influence of wind load distribution along the greenhouse longitudinal direction. The schematic diagram of the greenhouse skeleton is shown in Fig. 4(a), in which y and z were respectively along the transverse and vertical direction of the skeleton, and x was along the longitudinal direction of the greenhouse skeleton. The constitutive model of steel is shown in Fig. 4(b).

Fig. 4
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Steel skeleton and constitutive model overview.

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The skeleton foot was welded to the embedded flat-elliptical pipe along the longitudinal direction of the greenhouse and the ground anchor. Therefore, the structure had a large stiffness, which was defined as the fixed end, in which degrees of freedom in all directions were restrained. In this model, the boundary conditions at the base of the skeleton are assumed to be fully fixed. However, future studies may incorporate the effects of local flexibility and connection stiffness to refine the structural response18,19.

There were 7 longitudinal tie rods, which were connected to the skeleton by buckles.The longitudinal tie rods were bound to the arch rod. By inquiring with the manufacturer, it was found that all the skeletons were made with Q235 thin-wall galvanized steel pipe, the greenhouse film was made with Polyolefin. According to the Chinese cord, the material parameter of steel pipe and film are listed in Table 120.

Table 1 Material parameter of steel pipe and film.
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where: b is the width of section; h is the depth of section; t is thickness of section; ρ is density; fy is the yield strength of steel; λ is Poisson ratio of steel; E is elasticity modulus of steel.

The solid element C3D8R was used to build the flat-elliptical pipe skeleton. Compared to the beam elements commonly used by most researchers in modeling greenhouse skeletons, solid elements provide a more accurate representation of the stress distribution within the skeletons. This is particularly evident in the case of nonlinear analyses and dynamic simulations, where solid elements are better able to capture the vibrational characteristics of the skeletons. This enhanced accuracy leads to improved computational precision and more reliable analytical results.The C3D8R element is an 8-node hexahedral linear reduced integration element, with three degrees of freedom at each node. Compared to fully integrated elements, reduced integration elements contain only one integration point at the center of the element. The main advantages of this element include: effectively preventing shear locking under bending loads; providing high accuracy in displacement solutions; and having a minimal effect on analysis precision when the mesh undergoes distortion. Therefore, this element is suitable for structural analysis under complex deformation conditions.

The membrane element M3D4R was selected to build the plastic film attached to the greenhouse. The mesh division has a significant impact on the calculation results of the model. Taking the accuracy of results and computational efficiency into account, the steel skeleton was meshed by its size, and the approximate global size was set 2, and the number of elements per steel skeleton was 588. The greenhouse film was also meshed by size, the approximate global size was 1, and the overall number of elements of the greenhouse film was 1960. The tie rods were connected to the skeleton using a rigid connection, where all degrees of freedom are restrained. The film was connected to the skeleton at specific points using a binding connection to ensure that the film interacts with the skeleton as a unit.

According to the regulations 5.1.2 for the Technical specification for membrane structures(CECS 158:2015), the equilibrium shape of a membrane structure under prestress must be determined through an initial shape analysis21. This analysis serves as the basis for subsequent evaluation of structural performance under various load conditions. Therefore, prior to conducting any load effect analysis, it is essential to complete the initial shape analysis to ensure that the structural response is assessed based on a realistic and mechanically stable configuration. After conducting form-finding analysis on the film based on the nonlinear finite element method, the pre-stress of the film was determined to be 5 MPa for subsequent analysis.

In this study, the FEPG with 50 skeletons and films was established as shown in Fig. 5(a). The schematic diagram of 50 skeletons is shown in Fig. 5(b), in which the spacing of each skeleton was 1 m. The skeleton mesh is shown in Fig. 5(c).

Fig. 5
figure 5

Finite element model of the FEPG.

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As shown in Fig. 5(b), the No.1 skeleton was defined as the end skeleton according to the actual location in the project. So as the No.25 skeleton was defined as the center skeleton.

Analytical method of wind-induced vibration response

To consider the geometric nonlinearity of the FEPG under wind loads, the time-domain analysis method was adopted in this study, instead of frequency-domain analysis. The dynamic equilibrium equation for the FEPG under wind loads is

$$M\ddot {U}(t)+C\dot {U}(t)+KU(t)=F(t)$$
(1)

where M, C, K are the mass, damping, and stiffness matrices of the FEPG, \(\ddot {U}(t)\), \(\dot {U}(t)\), \(U(t)\) are the acceleration, velocity, and displacement vectors of the structural nodes; and \(F(t)\) is the wind load vector. The damping matrix is assumed to be given by the Rayleigh damping expression

$$C={a_M}M+{a_K}K$$
(2)

where \({a_M}\) and \({a_K}\)are the mass and stiffness proportional damping factors. These can be calculated as follows:

$${a_M}=2\xi {\omega _m}{\omega _n}/({\omega _m}+{\omega _n})$$
(3)
$${a_K}=2\xi /({\omega _m}+{\omega _n})$$
(4)

where \({\omega _m}\) and \({\omega _n}\) are the natural circular frequencies of the modes m and n, respectively. They are calculated by the modal analysis in Sect. Analytical method of wind-induced vibration response. \(\xi\) is the damping ratio. As there have been no reports on the measurement of damping ratio of the FEPG, \(\xi =0.02\) is assumed in this study, which is commonly used in steel structure design10.

Model analysis

The wind-induced vibration performance of structures is related to not only the fluctuating characteristics of incoming flow but also the dynamic characteristics of the FEPG, namely its modal frequencies and mode shapes. Hence, before analyzing wind-induced vibration response, the modal analysis of the FEPG should be first conducted using ABAQUS. In addition, the modal analysis is also an effective means of checking the finite element model. To calculate the mass proportional damping factor \({a_M}\) and stiffness proportional damping factor \({a_K}\) in Eq. (2), modal analyses must be performed before wind-induced dynamic response analyses. Based on the Lanczos method, the first 5 natural frequencies and vibration modes of the FEPG were calculated using ABAQUS software. Table 2 lists the first 5 natural frequencies of the FEPG. The first 5 vibration modes are presented in Fig. 6.

Table 2 First 5 natural frequencies and instability modes of the structure.
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Fig. 6
figure 6

The first 5 vibration modes of the structure.

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Calculation of wind loads

In this study, the mean wind speed was calculated conforming to the Chinese standard, and the autoregressive method in linear filtering were used to simulate the fluctuating wind speed2. The full dynamic wind speed applied to the greenhouse is obtained by putting the two winds together.

Calculation of mean wind speed

The plastic greenhouse calculated is located in Jiamusi, Heilongjiang Province, China. According to the Chinese standard, the design service life of the plastic greenhouse is 10 years, the basic wind pressure in Jiamusi is 0.38kN/m2, which corresponds to the instantaneous wind speed at 3-second time interval2. However, according to the Chinese standard, the basic wind pressure in Jiamusi is 0.65kN/m2 in 50-year recurrence at 10-minute time interval3. According to the Chinese standard2, the ratio of wind pressure in 10-year recurrence to wind pressure in 50-year recurrence is 0.734, therefore the basic wind pressure in 10-year recurrence at 10-minute interval is 0.65 × 0.734 = 0.477kN/m2. The basic wind pressure is 0.477kN/m2 for the sake of security.

According to the Bernoulli equation in fluid mechanics, the relationship between wind pressure and wind speed is:

$$w=\frac{1}{2}\rho {v^2}=\frac{\gamma }{{2g}}{v^2}$$
(5)

where w is wind pressure, kN/m2, \(\rho\) is air density, kPa, v is wind speed, m/s, \(\gamma\) is air gravity density, kN/m3, g is gravitation acceleration, m/s2.

In the above equation, the air gravity density \(\rho\) is 0.012018kN/m3 at the standard atmospheric pressure of 101.235 kPa. Jiamusi is located near 45° north latitude, g = 9.8 m/s2. Under these condition, the wind pressure can be calculated by Eq. (5):

$$w=\frac{1}{2}\rho {v^2}=\frac{{0.012018}}{{2 \times 9.8}}{v^2}=\frac{{{v^2}}}{{1630}}$$
(6)

According to relationship between wind pressure and wind speed, the mean wind speed is 27.88 m/s under the condition of 10-meter standard heigh, standard geomorphy B, 10-year recurrence and 10-minute interval.

Simulation of fluctuating wind speed

Compared with earthquake waves that can be measured in actual earthquake, wind speed time history acted on structures in practical engineering is not easy to measure, and at present there is no real data available for use. Therefore, Matlab software is used to simulate fluctuating wind speed as a random process22.The fluctuating wind speed can be expressed as a Gaussian stationary random process and simulated by using the linear filtering method23. Autoregressive(AR) method is a special linear filtering method, which has the main advantages of simple calculation, high accuracy and high convergence speed24. The selection of the autoregressive (AR) model in this study was based on its distinct advantages over the harmonic superposition method for simulating fluctuating wind speed time series. Specifically, the AR model offers superior flexibility in handling the inherent randomness, temporal dependencies, non-stationarity, and complex spectral characteristics of wind speed data. Its computational efficiency and adaptability make it a preferred approach for modern wind speed simulation, particularly when dealing with intricate and stochastic wind time series that require accurate representation of both short-term and long-term variability. So it is a good method for simulating the fluctuating wind speed time history. The basic idea of AR method is to predict the future using the previous data in time series.

The time history of fluctuating wind speed for M-points AR model in space can be expressed as follows12,13:

$$V(X,Y,Z,t)=\sum\limits_{{k=1}}^{p} {{\psi _k}} V(X,Y,Z,t{\text{-}}K\Delta t)+N(t)$$
(7)

where X = (xt,...,xm)T, Y = (yt,...,ym)T, Z = (zt,...,zm)T are the 3-D coordinates of M-points in space, V (X, Y, Z, t) is the \(M \times 1\) column vector, K is the coefficient related to the wind pressure distribution on the structure, p is the order of AR model, \(\Delta t\) is the time step size for simulating wind speed time history, \({\psi _k}\) is the autoregressive coefficient matrix of AR model, k = 1,…,p; \(N(t)\) is the variable in independent random process, which can be obtained from the following formula:

$$N(t)=L \cdot n(t)$$
(8)

where \(n(t)=[{n_1}(t),...,{n_m}(t)]\), \({n_i}(t)\), i = 1,...,M, is the independent Gaussian random process, in which mean is 0, variance is 1, L is the M-order lower triangular matrix, as shown in the following formula:

$$L=\left[ {\begin{array}{*{20}{c}} {{L_{11}}}&0&{\begin{array}{*{20}{c}} \cdots &0 \end{array}} \\ {{L_{12}}}&{{L_{22}}}&{\begin{array}{*{20}{c}} \cdots &0 \end{array}} \\ {{L_{M1}}}&{{L_{M2}}}&{\begin{array}{*{20}{c}} \cdots &{{L_{MM}}} \end{array}} \end{array}} \right]$$
(9)
$${L_{ij}}=\frac{{{R_{ij}} - \sum\limits_{{k=1}}^{{i - 1}} {{L_{ik}}{L_{jk}}} }}{{{L_{jj}}}},\,{L_{ii}}=\sqrt {{R_{ij}} - \sum\limits_{{k=1}}^{{i - 1}} {L_{{_{{ik}}}}^{2}} },\,i,j=1,2, \cdots M$$
(10)

The relationship between the covariance matrix R and the regression coefficients \(\psi\) of the random wind process can be written as follows:

(11)

where \(\psi ={\left[ {I,{\psi _1},...,{\psi _{\text{p}}}} \right]^T}\) is the \((p+1)M \times M\)-order matrix, \({O_P}\) is the \(p(M \times M)\)-order matrix, \({R_N}\) is the M-order covariance matrix, which can be decomposed using Cholesky decomposition as follows:

$${R_N}=L \cdot {L^T}$$
(12)

where R is the \((p+1)M \times (p+1)M\)-order autocorrelation Toeplitz matrix, which can be obtained as follows:

(13)

where \({R_{{\text{ij}}}}(m\Delta t)\) is the M * M-order matrix,\(i,j=1,...,p+1, i \ne j\),\(m=1,...,p\), which can be calculated from the power spectral density function using the Wiener-Khintchina formula as follows:

$${R_{{\text{ij}}}}(\tau )=\int_{0}^{\infty } {{S_v}(f)\cos (2\pi f \cdot \tau )} df, i,k=1,...,M$$
(14)

Before calculation, it is necessary to assume that the wind speed before the initial moment is 0, as \(t \leqslant 0\), \(V(t)=0\). Finally, the time history of fluctuating wind speed for M-points AR model can be demonstrated as follows:

$$\left[ {\begin{array}{*{20}{c}} {{v^1}(j\Delta t)} \\ \vdots \\ {{v^M}(j\Delta t)} \end{array}} \right]=\sum\limits_{{k=1}}^{p} {{\psi _k}} \cdot \left[ {\begin{array}{*{20}{c}} {{v^1}\left[ {(j - k)\Delta t} \right]} \\ \vdots \\ {{v^M}\left[ {(j - k)\Delta t} \right]} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {{n^1}(j\Delta t)} \\ \vdots \\ {{n^M}(j\Delta t)} \end{array}} \right]\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} {j\Delta t=0,...,T} \\ {k \leqslant j} \end{array}$$
(15)

The calculation results of Davenport fluctuating wind speed spectrum in high frequency band are relatively conservative, which makes the structure safer. Therefore, Davenport fluctuating wind power spectrum was used to simulate the fluctuating wind perpendicular to greenhouse surface, which can be expressed as:

$${S_v}{\text{(}}f{\text{)}}=\overline {V} _{{10}}^{2}\frac{{{\text{4}}K{x^{}}}}{{f{{{\text{(1}}+{x^{\text{2}}}{\text{)}}}^{{\text{4/3}}}}}}$$
(16)
$$x=1200\frac{f}{{\overline {V} _{{10}}^{{}}}}$$
(17)

where \({S_v}{\text{(}}f{\text{)}}\) is the auto-power of the fluctuating wind speed, f is the fluctuating wind speed frequency, K is the surface drag coefficient, \(\overline {V} _{{10}}^{{}}\) is the mean wind speed at 10-meter height with 10-minute interval, x is the integral scale coefficient of turbulence, which is 0.02 in Davenport fluctuating wind speed spectrum.

The correlation coefficient represents the likelihood that when the fluctuating wind speed at a specific point in space reaches its maximum value, other points within the same space do not necessarily reach their maximum simultaneously. The closer the points are, the higher the probability of them reaching the maximum value together. Therefore, for large structures with extensive wind-exposed areas and spans, it is essential to consider the spatial correlation of wind fluctuations and the correlation coefficient. Correlation functions can be broadly categorized into two types: the first type uses both position and frequency as independent variables, while the second type only considers position as an independent variable.

Referring to the experience and methods of scholars at home and abroad in simulating the fluctuating wind speed time history curve of large-span structures, combined with the time-domain analysis method, only the position was selected as the independent variable because of the small span of the FEPG. The correlation along longitudinal (x-direction) and vertical (z-direction) directions of the greenhouse were considered. Shiotani correlation coefficient was used to simulate the time history curve of fluctuating wind speed, which can be expressed as:

$${\text{coh}}(x,x') ={\text{exp}}( - \frac{{\left| {x - x'} \right|}}{{{L_x}}}),{\text{coh}} (z,z') ={\text{exp}}( - \frac{{\left| {z - z'} \right|}}{{{L_z}}})$$
(18)

where \({L_x}\) is the integration dimension of turbulent flow in x-direction, which is 50, \({L_z}\) is the integration dimension of turbulent flow in z-direction, which is 60.

Based on the experience and conclusions of previous scholars and a number of attempts on different parameters, the simulation parameters for the fluctuating wind speed are as follows:

  1. (1)

    The mean wind speed is 27.88 m/s at 10-meter height in 10-year recurrence in Jiamusi, Heilongjiang Province;

(2) The autoregressive order is 4;

(3) The total duration is 60 s and the time step is 0.1.

(4) The upper cut-off frequency is 10 Hz and the frequency step is 0.01 Hz;

(5) The surface drag coefficient K is 0.003;

The influence of the correlation coefficient on the simulated fluctuating wind speed is reflected in the relationship of the distance and wind speed between the calculation location and the center skeleton. That is, the maximum fluctuating wind speed is applied to the center skeleton, the farther from the center skeleton, the weaker the correlation is and the smaller wind speed becomes. The time history curves of fluctuating wind speed at the top of the center skeleton (No.25) and the end skeleton (No.1) are shown in Fig. 7.

Fig. 7
figure 7

The time history curves of fluctuating wind speed at the top of skeleton.

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The simulated wind speed power spectra at the top of the skeleton with Davenport spectrum are shown in Fig. 8. It can be seen that the simulated wind speed power spectra agreed well with the Davenport spectrum, which indicates that the simulated wind speeds were reliable.

Fig. 8
figure 8

Comparison of the simulated spectrum and Davenport spectrum at the top of skeleton.

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Calculation of full dynamic wind loads

After the fluctuating wind speed was calculated, the full dynamic wind speed was obtained by adding the fluctuating wind speed to the mean wind speed. According to the Bernoulli equation, it can be transformed into the full dynamic wind pressure acting on the structure.

Conforming to the Chinese standard, for the main structure of the greenhouse, the calculation formula of wind load perpendicular to the greenhouse surface is2:

$${w_k}={\mu _{\text{s}}}{\mu _{\text{z}}}w$$
(19)

Where \({w_k}\) is the standard value of the wind load (kN m−2); \({\mu _{\text{s}}}\) is the shape coefficient of wind pressure shown in Fig. 9; \({\mu _{\text{z}}}\) is the wind pressure height variation coefficient; w is the wind pressure acting on structure (kN m−2), and here it is the full dynamic wind pressure.

Fig. 9
figure 9

Shape coefficient of wind load on the FEPG.

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The time history curves of full dynamic wind load at the top of the center skeleton and the end skeleton are shown in Fig. 10.

Fig. 10
figure 10

The time history curves of full dynamic wind load at the top of skeleton.

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In this study, the full dynamic wind load time history curve was loaded on the film of the FEPG in the form of surface load. Then the wind load was transferred from the greenhouse film to the greenhouse skeleton.

Results and discussion

Model validation

Chai performed numerical analysis on a skeleton of single arch greenhouse subjected to vertical loads25. The model similarity ratio was 1:4. The load and global geometric parameters of the single-span vaulted-roof greenhouse are shown in Fig. 11. The greenhouse skeleton was made of circular steel pipe (\(\phi\)8 mm × 0.5 mm) and the type of steel was Q345. The constitutive relationship was an ideal Elastic-plastic model. Both ends of the greenhouse were fixed, and the vertical loads were applied on the finite element model as concentrated force.

Fig. 11
figure 11

Load and global geometric parameters of the single-span vaulted-roof greenhouse.

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In this study, the C3D8R solid element was used to simulate the greenhouse skeleton in ABAQUS finite element software. While in the original literature, the ANSYS software was used to analyze the skeleton.

The stress diagrams of the greenhouse skeleton by using the two software are shown in in Fig. 12. The figures in Fig. 12 are stress at the same position by the two software.

Fig. 12
figure 12

Comparison of the stress diagrams of the greenhouse.

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From Fig. 12, it can be seen that the stress calculated by the proposed method is basically consistent with that in the literature. The maximum stress in literature is 345.25 MPa, while it is 331.3 MPa in the present result, and the relative errors is 4.04%. The stress error range of all calculation points are 3.56%−4.51%, which proved the accuracy of the finite element modelling method. The boundary conditions, member geometry, and material properties applied in this study are consistent with those reported in the literature.The primary factor contributing to the differences in stress results is the use of the BEAM188 beam element in the literature for modeling the skeleton, while this study employs the C3D8R solid element for the same purpose. The variation in element type and meshing strategies leads to differing stress distributions between the two models.

While the model has been validated under static loading conditions, dynamic validation is an important next step. Future research will include wind tunnel tests and field monitoring to provide a more accurate validation of the numerical model.

Deformation analysis

In order to investigate whether the displacement of the greenhouse skeleton satisfy the safety design requirements or not, and make clear the deformation of the flat-elliptical pipe plastic greenhouse skeleton subjected to the full dynamic wind load and the mean wind load, displacement diagrams of the center skeleton at the moment of maximum displacement subjected to full dynamic and mean wind load are extracted respectively, as shown in Fig. 13. The displacement diagrams in this study were magnified 50 times for better observation.

Fig. 13
figure 13

Displacement diagrams of the center skeleton of the greenhouse subjected to different loads.

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As shown in Fig. 13(a), under the full dynamic wind load, the maximum displacement of the greenhouse skeleton is 10.49 mm at the shoulder of the leeward side. From Fig. 13(b), it can be seen that under the mean wind load, the maximum displacement of the greenhouse skeleton is 4.08 mm at the shoulder of the leeward side. The center skeleton was divided into 10 sections along the span direction, and the displacement of center node at each section in Fig. 13 are shown in Fig. 14.

Fig. 14
figure 14

Comparison of the vector displacement of the greenhouse skeleton subjected to full dynamic wind load and mean wind load.

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As shown in Fig. 14, the skeleton vector displacement trend subjected to full dynamic wind load and mean wind load is similar, which shows a bimodal asymmetric distribution. The shoulder on the windward side is concave while the shoulder on the leeward side is convex.The maximum vector displacement both occurred in the shoulder area of the leeward side. From Fig. 14, it can also be seen that the displacement of the center skeleton subjected to full dynamic wind load was about 2.57 times that of that subjected to mean wind load. Li concluded that the ratio was 2.02, which was 27.2% larger than that in this study13. The rectangular pipe skeleton solar greenhouse was investigated by Li, which was different from the stiffness of the FEPG in this study. The greenhouse skeleton is very sensitive to wind, so the displacement of the greenhouse skeletons is much greater after the fluctuation of wind is considered. That is why greenhouse skeletons often fails subjected to strong winds which is safe enough conforming to the design code.

To further investigate the variation law of greenhouse skeleton displacement subjected to full dynamic wind load, displacement and the full dynamic wind load time history curves of the maximum displacement node are obtained and shown in Fig. 15. Figure 15(a) is for the fourth point from the left in Figs. 14 and 15(b) is for the seventh point from the left in Fig. 14.

Fig. 15
figure 15

Displacement and the full dynamic wind load time history curves of the maximum displacement node.

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According to the Chinese standard, the maximum allowable displacement of the skeleton should not exceed 24 mm, and the maximum allowable stress in the skeleton should be 235MPa19. These limits provide a clear benchmark for evaluating the structural safety of the greenhouse under wind loads.

From Fig. 15, it can be seen that the blue line in Fig. 15(a) is the same with Fig. 15(b) because the variation of wind along the span direction of greenhouse skeleton was not considered in the full dynamic wind load. The whole trend of vibration displacement time history curve is similar to the full dynamic wind load time history curve on both the windward and the leeward. But the vibration displacement and wind load don’t reach the maximum at the same time, and there are lags and leads. The maximum vibration displacement is 9.05 mm on the windward side occurs at 27.5 s, while it is 10.49 mm on the leeward side at 19.97 s. At the same time, it can be seen that the vibration is steady than the wind load, that is the vibration frequency of structure is less than the wind load vibration frequency. The observed phenomenon of vibration frequencies being lower than the wind load frequency can be explained by the resonance effect and damping characteristics of the structure. The structural damping plays a key role in modulating the dynamic response, and future studies could include damping coefficients to better capture this interaction.

As shown in Fig. 15, in the first 2.5s, the vibration is intense and the vibration displacement amplitude is greater. Then it gradually vibrates with the wind load at the same frequency. The reason is that the initial vibration comprises the free vibration and the forced vibration caused by full dynamic wind load. When the free vibration attenuates, only forced vibration remains. Therefore, the vibration was similar to the change of wind load after 2.5s. This phenomenon accords with the basic principle of forced vibration in structural dynamics.

The above greenhouse skeleton displacement is the total vector displacement of the node composing the horizontal and vertical displacement vectors. But the control displacements given in the codes are the displacement along the coordinate axis in general. Therefore, the maximum node displacement on the windward and leeward of the center skeleton is decomposed the horizontal displacement along the y axis and vertical displacement along the z axis as shown in Fig. 16.

Fig. 16
figure 16

Displacement and wind load time history of node with the maximum displacement on leeward.

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From Fig. 16(a), on the windward, both the maximum horizontal displacement 7.82 mm and the vertical displacement − 4.55 mm occur at 27.5 s. The horizontal displacement is to the right and the vertical displacement was downward. The windward skeleton is concave. From Fig. 16(b), on the leeward, both the maximum horizontal displacement 7.82 mm and the vertical displacement − 4.55 mm occur at 27.45 s. The horizontal displacement is to the right and the vertical displacement direction was upward. The windward skeleton is convex. According to the Chinese standard, the nodes displacement satisfy the safety requirements of the skeleton during the elastic stage19.

Stress analysis

To find out the Mises stress of the FEPG skeleton subjected to full dynamic wind load and mean wind load, the stress distribution of greenhouse skeleton was analyzed. The skeleton was divided into 10 sections along the span direction. The nodes’ Mises stress in the center skeleton subjected to full dynamic wind load and mean wind load at the moment of the maximum Mises stress are obtained respectively shown in Fig. 17.

Fig. 17
figure 17

Comparison of the Mises stress of the greenhouse skeleton nodes subjected to full dynamic wind load and mean wind load.

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As shown in Fig. 17, the maximum Mises stress of the skeleton 26.19 MPa subjected to full dynamic wind and 17.01 MPa for load and mean wind load both appear in the windward foot area. The minimum stress occurs at the leeward foot area.

The Mises stress of skeleton shows a decrease tendency from the windward foot to the windward waist and it is a bimodal asymmetric distribution in the windward waist and the top of the skeleton. The maximum Mises stress of the center skeleton subjected to the full dynamic wind load was about 1.53 times that subjected to the mean wind load. Compared with Wang, the result in this study is 23.5% greater11.

To explore the variation law of Mises stress of greenhouse skeleton subjected to full dynamic wind load, Mises stress and the full dynamic wind load time history curve of the maximum Mises stress node on the windward side and leeward side are shown in Fig. 18.

Fig. 18
figure 18

Mises stress and the full dynamic wind load time history curves of the maximum Mises stress node.

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From Fig. 18, it can be seen that the variation law of stress and displacement of greenhouse skeleton subjected to the full dynamic wind load is similar. The time history curve of node vibration Mises stress and the full dynamic wind load exhibit similar trends, but they don’t reach the their peaks at the same time. The maximum vibration Mises stress is 26.19 MPa on the windward side occurs at 36.55 s, while it is 17.54 MPa on the leeward side occurs at 19.97 s. Meanwhile, the vibration frequency of structure is less than the wind load vibration frequency. The initial section of the stress time history curve also contains free vibration. Only after the free vibration attenuates, the node vibration vibrates with the full dynamic wind load.

According to the Chinese standard, the nodes stress satisfy the safety requirement at the elastic stage of the skeleton17.

Longitudinal displacement and stress of greenhouse skeletons

To explore the variation law of different skeletons’ vector displacement distribution subjected to the full dynamic wind load and mean wind load, the different skeleton nodes’ vector displacement curves alone the longitudinal direction are obtained at the moment when center skeleton reached the maximum displacement, shown in Fig. 19.

The legend in the figure from top to bottom successively represent the nodes’ displacement curve on the center skeleton (No.25)、No.24、No.20、No.15、No.10、No.5、the end skeleton (No.1) and the nodes’ displacement curves of the center skeleton subjected to the mean wind load at the moment when the center skeleton reaches the maximum displacement.

Fig. 19
figure 19

Comparison of the vector displacement of the greenhouse skeletons subjected to the full dynamic wind load and mean wind load.

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From Fig. 19, it can be seen that when the center skeleton reaches the maximum displacement, the displacement of adjacent skeleton (No.24) is slightly smaller than that of the center skeleton. Meanwhile, the displacement of other skeleton decrease with their distance from the center skeleton increase. These results confirm the correlation law of the fluctuating wind field.

That is, there exists a cross-correlation function between different nodes in the wind field. The maximum value of the cross-correlation function represents the degree of correlation between the two nodes. The closer the distance between two nodes, the larger the correlation function value and the stronger the correlation is. This study loaded the maximum full dynamic wind load on the center skeleton, therefore the further distance from the center skeleton, the smaller wind load on the skeleton, and the smaller displacement on the skeleton24. The minimum vector displacement of the end skeleton (No.1) subjected to the full dynamic wind load is 1.05 times that subjected to the mean wind load.

To investigate the variation law of different skeletons’ Mises stress subjected to the full dynamic wind load and the mean wind load, the different skeleton nodes’ Mises stress curves are obtained at the moment when the center skeleton reaches the maximum stress, as shown in Fig. 20.

Fig. 20
figure 20

Comparison of the Mises stress of the greenhouse skeletons subjected to the full dynamic wind load and mean wind load.

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Compared Fig. 20 with Fig. 19, it can be seen that Mises stress and the displacement of the greenhouse skeleton along the longitudinal direction is similar, and the displacement decreases with the distance from the center skeleton increase. The minimum Mises stress of end skeleton (No.1) subjected to the full dynamic wind load is about 1.02 times that subjected to the mean wind load.

Conclusions

The FEM model of the FEPG with film was established using ABAQUS software. Fluctuating wind speeds were simulated by using the Linear Filtering Method, and the full dynamic wind speed was obtained by adding the fluctuating wind speed to the mean wind speed together. The greenhouse model was analyzed under both full dynamic and mean wind loads, considering fluctuating wind in the wind-induced response analysis. The main findings are as follows:

  1. (1)

    After the effects of fluctuating wind was considered, the load distribution on the greenhouse skeleton is non-uniform. Displacement is higher at the shoulders (windward and leeward sides) and lower at the feet. The stress distribution on the skeleton is asymmetric bimodal on both shoulders, with higher stress on the windward foot.

  2. (2)

    After the effects of fluctuating wind was considered, for the skeleton bearing the maximum force, the maximum displacement subjected to the full dynamic wind is 2.57 time of the displacement subjected to the mean wind, and the maximum displacement subjected to the full dynamic wind is 1.53 time of the displacement subjected to the mean wind.

  3. (3)

    After the effects of fluctuating wind was considered, for the skeleton bearing the minimum force, the maximum displacement and Mises stress subjected to the full dynamic wind are 1.05 and 1.02 times that subjected to the mean wind loads, respectively.

The findings of this study reveal the significant impact of fluctuating wind on the displacement and stress distribution of FEPGs, providing new insights into the dynamic behavior of greenhouse structures. These insights highlight the need for incorporating dynamic wind loads into the design to prevent failures, a consideration that is often ignored in current design codes. Plastic greenhouses are light structure and sensitive to wind, therefore, it is suggested that the adverse effects of fluctuating wind on the structure should be considered in design so as to enhance the resistance of this kind of structure subjected to wind load.

This study provides a fundamental analysis of the wind-induced responses of flat-elliptical pipe skeleton greenhouses, emphasizing the importance of considering fluctuating wind in structural design. The results can serve as a reference for the design and optimization of agricultural greenhouses, particularly in regions subject to high wind risks.

In future studies, it is essential to further investigate failure mechanisms such as fatigue accumulation, local buckling of structural elements, and connection failure under dynamic wind loads26. These factors could significantly affect the long-term performance and safety of the greenhouse structure. While support stiffness, initial imperfections, and second-order effects are not considered in this study, future research could integrate these factors to provide a more comprehensive understanding of the structural behavior under extreme wind loads. The impact of these assumptions on the overall results should be analyzed in future studies to refine the model.